YES(O(1),O(n^1)) 79.79/20.66 YES(O(1),O(n^1)) 79.79/20.66 79.79/20.66 We are left with following problem, upon which TcT provides the 79.79/20.66 certificate YES(O(1),O(n^1)). 79.79/20.66 79.79/20.66 Strict Trs: 79.79/20.66 { lt0(x, Nil()) -> False() 79.79/20.66 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.66 , lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.66 , g(x, Nil()) -> 79.79/20.66 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.66 , g(x, Cons(x', xs)) -> 79.79/20.66 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.66 , f(x, Nil()) -> 79.79/20.66 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.66 , f(x, Cons(x', xs)) -> 79.79/20.66 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.66 , number4(n) -> 79.79/20.66 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.66 , notEmpty(Nil()) -> False() 79.79/20.66 , notEmpty(Cons(x, xs)) -> True() 79.79/20.66 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.66 Weak Trs: 79.79/20.66 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.66 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.66 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.66 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.66 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.66 g(xs, Cons(Cons(Nil(), Nil()), y)) } 79.79/20.66 Obligation: 79.79/20.66 innermost runtime complexity 79.79/20.66 Answer: 79.79/20.66 YES(O(1),O(n^1)) 79.79/20.66 79.79/20.66 The weightgap principle applies (using the following nonconstant 79.79/20.66 growth matrix-interpretation) 79.79/20.66 79.79/20.66 The following argument positions are usable: 79.79/20.66 Uargs(f[Ite][False][Ite]) = {1}, Uargs(g[Ite][False][Ite]) = {1}, 79.79/20.66 Uargs(Cons) = {1, 2} 79.79/20.66 79.79/20.66 TcT has computed the following matrix interpretation satisfying 79.79/20.66 not(EDA) and not(IDA(1)). 79.79/20.66 79.79/20.66 [f[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [0] 79.79/20.66 79.79/20.66 [True] = [4] 79.79/20.66 79.79/20.66 [Nil] = [0] 79.79/20.66 79.79/20.66 [lt0](x1, x2) = [0] 79.79/20.66 79.79/20.66 [g](x1, x2) = [0] 79.79/20.66 79.79/20.66 [f](x1, x2) = [0] 79.79/20.66 79.79/20.66 [g[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [0] 79.79/20.66 79.79/20.66 [Cons](x1, x2) = [1] x1 + [1] x2 + [0] 79.79/20.66 79.79/20.66 [number4](x1) = [1] x1 + [7] 79.79/20.66 79.79/20.66 [notEmpty](x1) = [0] 79.79/20.66 79.79/20.66 [goal](x1, x2) = [1] x1 + [1] x2 + [7] 79.79/20.66 79.79/20.66 [False] = [0] 79.79/20.66 79.79/20.66 The order satisfies the following ordering constraints: 79.79/20.66 79.79/20.66 [f[Ite][False][Ite](True(), x', Cons(x, xs))] = [4] 79.79/20.66 > [0] 79.79/20.66 = [f(x', xs)] 79.79/20.66 79.79/20.66 [f[Ite][False][Ite](False(), Cons(x, xs), y)] = [0] 79.79/20.66 >= [0] 79.79/20.66 = [f(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.66 79.79/20.66 [lt0(x, Nil())] = [0] 79.79/20.66 >= [0] 79.79/20.66 = [False()] 79.79/20.66 79.79/20.66 [lt0(Nil(), Cons(x', xs))] = [0] 79.79/20.66 ? [4] 79.79/20.66 = [True()] 79.79/20.66 79.79/20.66 [lt0(Cons(x', xs'), Cons(x, xs))] = [0] 79.79/20.66 >= [0] 79.79/20.66 = [lt0(xs', xs)] 79.79/20.66 79.79/20.66 [g(x, Nil())] = [0] 79.79/20.66 >= [0] 79.79/20.66 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.66 79.79/20.66 [g(x, Cons(x', xs))] = [0] 79.79/20.66 >= [0] 79.79/20.66 = [g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.66 79.79/20.66 [f(x, Nil())] = [0] 79.79/20.66 >= [0] 79.79/20.66 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.66 79.79/20.66 [f(x, Cons(x', xs))] = [0] 79.79/20.66 >= [0] 79.79/20.66 = [f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.66 79.79/20.66 [g[Ite][False][Ite](True(), x', Cons(x, xs))] = [4] 79.79/20.66 > [0] 79.79/20.66 = [g(x', xs)] 79.79/20.66 79.79/20.66 [g[Ite][False][Ite](False(), Cons(x, xs), y)] = [0] 79.79/20.66 >= [0] 79.79/20.66 = [g(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.66 79.79/20.66 [number4(n)] = [1] n + [7] 79.79/20.66 > [0] 79.79/20.66 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.66 79.79/20.66 [notEmpty(Nil())] = [0] 79.79/20.66 >= [0] 79.79/20.66 = [False()] 79.79/20.66 79.79/20.66 [notEmpty(Cons(x, xs))] = [0] 79.79/20.66 ? [4] 79.79/20.66 = [True()] 79.79/20.66 79.79/20.66 [goal(x, y)] = [1] x + [1] y + [7] 79.79/20.66 > [0] 79.79/20.66 = [Cons(f(x, y), Cons(g(x, y), Nil()))] 79.79/20.66 79.79/20.66 79.79/20.66 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 79.79/20.66 79.79/20.66 We are left with following problem, upon which TcT provides the 79.79/20.66 certificate YES(O(1),O(n^1)). 79.79/20.66 79.79/20.66 Strict Trs: 79.79/20.67 { lt0(x, Nil()) -> False() 79.79/20.67 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.67 , lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.67 , g(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , g(x, Cons(x', xs)) -> 79.79/20.67 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.67 , f(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , f(x, Cons(x', xs)) -> 79.79/20.67 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.67 , notEmpty(Nil()) -> False() 79.79/20.67 , notEmpty(Cons(x, xs)) -> True() } 79.79/20.67 Weak Trs: 79.79/20.67 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.67 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.67 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 g(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , number4(n) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.67 Obligation: 79.79/20.67 innermost runtime complexity 79.79/20.67 Answer: 79.79/20.67 YES(O(1),O(n^1)) 79.79/20.67 79.79/20.67 The weightgap principle applies (using the following nonconstant 79.79/20.67 growth matrix-interpretation) 79.79/20.67 79.79/20.67 The following argument positions are usable: 79.79/20.67 Uargs(f[Ite][False][Ite]) = {1}, Uargs(g[Ite][False][Ite]) = {1}, 79.79/20.67 Uargs(Cons) = {1, 2} 79.79/20.67 79.79/20.67 TcT has computed the following matrix interpretation satisfying 79.79/20.67 not(EDA) and not(IDA(1)). 79.79/20.67 79.79/20.67 [f[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [True] = [0] 79.79/20.67 79.79/20.67 [Nil] = [0] 79.79/20.67 79.79/20.67 [lt0](x1, x2) = [1] 79.79/20.67 79.79/20.67 [g](x1, x2) = [0] 79.79/20.67 79.79/20.67 [f](x1, x2) = [1] x1 + [0] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [0] 79.79/20.67 79.79/20.67 [Cons](x1, x2) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [number4](x1) = [1] x1 + [7] 79.79/20.67 79.79/20.67 [notEmpty](x1) = [0] 79.79/20.67 79.79/20.67 [goal](x1, x2) = [1] x1 + [1] x2 + [7] 79.79/20.67 79.79/20.67 [False] = [4] 79.79/20.67 79.79/20.67 The order satisfies the following ordering constraints: 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](True(), x', Cons(x, xs))] = [1] x' + [0] 79.79/20.67 >= [1] x' + [0] 79.79/20.67 = [f(x', xs)] 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](False(), Cons(x, xs), y)] = [1] xs + [1] x + [4] 79.79/20.67 > [1] xs + [0] 79.79/20.67 = [f(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.67 79.79/20.67 [lt0(x, Nil())] = [1] 79.79/20.67 ? [4] 79.79/20.67 = [False()] 79.79/20.67 79.79/20.67 [lt0(Nil(), Cons(x', xs))] = [1] 79.79/20.67 > [0] 79.79/20.67 = [True()] 79.79/20.67 79.79/20.67 [lt0(Cons(x', xs'), Cons(x, xs))] = [1] 79.79/20.67 >= [1] 79.79/20.67 = [lt0(xs', xs)] 79.79/20.67 79.79/20.67 [g(x, Nil())] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [g(x, Cons(x', xs))] = [0] 79.79/20.67 ? [1] 79.79/20.67 = [g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.67 79.79/20.67 [f(x, Nil())] = [1] x + [0] 79.79/20.67 >= [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [f(x, Cons(x', xs))] = [1] x + [0] 79.79/20.67 ? [1] x + [1] 79.79/20.67 = [f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite](True(), x', Cons(x, xs))] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [g(x', xs)] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite](False(), Cons(x, xs), y)] = [4] 79.79/20.67 > [0] 79.79/20.67 = [g(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.67 79.79/20.67 [number4(n)] = [1] n + [7] 79.79/20.67 > [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [notEmpty(Nil())] = [0] 79.79/20.67 ? [4] 79.79/20.67 = [False()] 79.79/20.67 79.79/20.67 [notEmpty(Cons(x, xs))] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [True()] 79.79/20.67 79.79/20.67 [goal(x, y)] = [1] x + [1] y + [7] 79.79/20.67 > [1] x + [0] 79.79/20.67 = [Cons(f(x, y), Cons(g(x, y), Nil()))] 79.79/20.67 79.79/20.67 79.79/20.67 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 79.79/20.67 79.79/20.67 We are left with following problem, upon which TcT provides the 79.79/20.67 certificate YES(O(1),O(n^1)). 79.79/20.67 79.79/20.67 Strict Trs: 79.79/20.67 { lt0(x, Nil()) -> False() 79.79/20.67 , lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.67 , g(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , g(x, Cons(x', xs)) -> 79.79/20.67 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.67 , f(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , f(x, Cons(x', xs)) -> 79.79/20.67 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.67 , notEmpty(Nil()) -> False() 79.79/20.67 , notEmpty(Cons(x, xs)) -> True() } 79.79/20.67 Weak Trs: 79.79/20.67 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.67 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.67 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.67 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 g(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , number4(n) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.67 Obligation: 79.79/20.67 innermost runtime complexity 79.79/20.67 Answer: 79.79/20.67 YES(O(1),O(n^1)) 79.79/20.67 79.79/20.67 The weightgap principle applies (using the following nonconstant 79.79/20.67 growth matrix-interpretation) 79.79/20.67 79.79/20.67 The following argument positions are usable: 79.79/20.67 Uargs(f[Ite][False][Ite]) = {1}, Uargs(g[Ite][False][Ite]) = {1}, 79.79/20.67 Uargs(Cons) = {1, 2} 79.79/20.67 79.79/20.67 TcT has computed the following matrix interpretation satisfying 79.79/20.67 not(EDA) and not(IDA(1)). 79.79/20.67 79.79/20.67 [f[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [True] = [0] 79.79/20.67 79.79/20.67 [Nil] = [0] 79.79/20.67 79.79/20.67 [lt0](x1, x2) = [1] 79.79/20.67 79.79/20.67 [g](x1, x2) = [0] 79.79/20.67 79.79/20.67 [f](x1, x2) = [1] x1 + [0] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [0] 79.79/20.67 79.79/20.67 [Cons](x1, x2) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [number4](x1) = [1] x1 + [7] 79.79/20.67 79.79/20.67 [notEmpty](x1) = [0] 79.79/20.67 79.79/20.67 [goal](x1, x2) = [1] x1 + [1] x2 + [7] 79.79/20.67 79.79/20.67 [False] = [0] 79.79/20.67 79.79/20.67 The order satisfies the following ordering constraints: 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](True(), x', Cons(x, xs))] = [1] x' + [0] 79.79/20.67 >= [1] x' + [0] 79.79/20.67 = [f(x', xs)] 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](False(), Cons(x, xs), y)] = [1] xs + [1] x + [0] 79.79/20.67 >= [1] xs + [0] 79.79/20.67 = [f(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.67 79.79/20.67 [lt0(x, Nil())] = [1] 79.79/20.67 > [0] 79.79/20.67 = [False()] 79.79/20.67 79.79/20.67 [lt0(Nil(), Cons(x', xs))] = [1] 79.79/20.67 > [0] 79.79/20.67 = [True()] 79.79/20.67 79.79/20.67 [lt0(Cons(x', xs'), Cons(x, xs))] = [1] 79.79/20.67 >= [1] 79.79/20.67 = [lt0(xs', xs)] 79.79/20.67 79.79/20.67 [g(x, Nil())] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [g(x, Cons(x', xs))] = [0] 79.79/20.67 ? [1] 79.79/20.67 = [g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.67 79.79/20.67 [f(x, Nil())] = [1] x + [0] 79.79/20.67 >= [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [f(x, Cons(x', xs))] = [1] x + [0] 79.79/20.67 ? [1] x + [1] 79.79/20.67 = [f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite](True(), x', Cons(x, xs))] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [g(x', xs)] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite](False(), Cons(x, xs), y)] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [g(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.67 79.79/20.67 [number4(n)] = [1] n + [7] 79.79/20.67 > [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [notEmpty(Nil())] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [False()] 79.79/20.67 79.79/20.67 [notEmpty(Cons(x, xs))] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [True()] 79.79/20.67 79.79/20.67 [goal(x, y)] = [1] x + [1] y + [7] 79.79/20.67 > [1] x + [0] 79.79/20.67 = [Cons(f(x, y), Cons(g(x, y), Nil()))] 79.79/20.67 79.79/20.67 79.79/20.67 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 79.79/20.67 79.79/20.67 We are left with following problem, upon which TcT provides the 79.79/20.67 certificate YES(O(1),O(n^1)). 79.79/20.67 79.79/20.67 Strict Trs: 79.79/20.67 { lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.67 , g(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , g(x, Cons(x', xs)) -> 79.79/20.67 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.67 , f(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , f(x, Cons(x', xs)) -> 79.79/20.67 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.67 , notEmpty(Nil()) -> False() 79.79/20.67 , notEmpty(Cons(x, xs)) -> True() } 79.79/20.67 Weak Trs: 79.79/20.67 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.67 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , lt0(x, Nil()) -> False() 79.79/20.67 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.67 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.67 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 g(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , number4(n) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.67 Obligation: 79.79/20.67 innermost runtime complexity 79.79/20.67 Answer: 79.79/20.67 YES(O(1),O(n^1)) 79.79/20.67 79.79/20.67 The weightgap principle applies (using the following nonconstant 79.79/20.67 growth matrix-interpretation) 79.79/20.67 79.79/20.67 The following argument positions are usable: 79.79/20.67 Uargs(f[Ite][False][Ite]) = {1}, Uargs(g[Ite][False][Ite]) = {1}, 79.79/20.67 Uargs(Cons) = {1, 2} 79.79/20.67 79.79/20.67 TcT has computed the following matrix interpretation satisfying 79.79/20.67 not(EDA) and not(IDA(1)). 79.79/20.67 79.79/20.67 [f[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [True] = [0] 79.79/20.67 79.79/20.67 [Nil] = [0] 79.79/20.67 79.79/20.67 [lt0](x1, x2) = [0] 79.79/20.67 79.79/20.67 [g](x1, x2) = [0] 79.79/20.67 79.79/20.67 [f](x1, x2) = [1] x1 + [0] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [0] 79.79/20.67 79.79/20.67 [Cons](x1, x2) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [number4](x1) = [1] x1 + [7] 79.79/20.67 79.79/20.67 [notEmpty](x1) = [4] 79.79/20.67 79.79/20.67 [goal](x1, x2) = [1] x1 + [1] x2 + [7] 79.79/20.67 79.79/20.67 [False] = [0] 79.79/20.67 79.79/20.67 The order satisfies the following ordering constraints: 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](True(), x', Cons(x, xs))] = [1] x' + [0] 79.79/20.67 >= [1] x' + [0] 79.79/20.67 = [f(x', xs)] 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](False(), Cons(x, xs), y)] = [1] xs + [1] x + [0] 79.79/20.67 >= [1] xs + [0] 79.79/20.67 = [f(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.67 79.79/20.67 [lt0(x, Nil())] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [False()] 79.79/20.67 79.79/20.67 [lt0(Nil(), Cons(x', xs))] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [True()] 79.79/20.67 79.79/20.67 [lt0(Cons(x', xs'), Cons(x, xs))] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [lt0(xs', xs)] 79.79/20.67 79.79/20.67 [g(x, Nil())] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [g(x, Cons(x', xs))] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.67 79.79/20.67 [f(x, Nil())] = [1] x + [0] 79.79/20.67 >= [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [f(x, Cons(x', xs))] = [1] x + [0] 79.79/20.67 >= [1] x + [0] 79.79/20.67 = [f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite](True(), x', Cons(x, xs))] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [g(x', xs)] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite](False(), Cons(x, xs), y)] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [g(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.67 79.79/20.67 [number4(n)] = [1] n + [7] 79.79/20.67 > [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [notEmpty(Nil())] = [4] 79.79/20.67 > [0] 79.79/20.67 = [False()] 79.79/20.67 79.79/20.67 [notEmpty(Cons(x, xs))] = [4] 79.79/20.67 > [0] 79.79/20.67 = [True()] 79.79/20.67 79.79/20.67 [goal(x, y)] = [1] x + [1] y + [7] 79.79/20.67 > [1] x + [0] 79.79/20.67 = [Cons(f(x, y), Cons(g(x, y), Nil()))] 79.79/20.67 79.79/20.67 79.79/20.67 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 79.79/20.67 79.79/20.67 We are left with following problem, upon which TcT provides the 79.79/20.67 certificate YES(O(1),O(n^1)). 79.79/20.67 79.79/20.67 Strict Trs: 79.79/20.67 { lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.67 , g(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , g(x, Cons(x', xs)) -> 79.79/20.67 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.67 , f(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , f(x, Cons(x', xs)) -> 79.79/20.67 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) } 79.79/20.67 Weak Trs: 79.79/20.67 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.67 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , lt0(x, Nil()) -> False() 79.79/20.67 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.67 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.67 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 g(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , number4(n) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , notEmpty(Nil()) -> False() 79.79/20.67 , notEmpty(Cons(x, xs)) -> True() 79.79/20.67 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.67 Obligation: 79.79/20.67 innermost runtime complexity 79.79/20.67 Answer: 79.79/20.67 YES(O(1),O(n^1)) 79.79/20.67 79.79/20.67 The weightgap principle applies (using the following nonconstant 79.79/20.67 growth matrix-interpretation) 79.79/20.67 79.79/20.67 The following argument positions are usable: 79.79/20.67 Uargs(f[Ite][False][Ite]) = {1}, Uargs(g[Ite][False][Ite]) = {1}, 79.79/20.67 Uargs(Cons) = {1, 2} 79.79/20.67 79.79/20.67 TcT has computed the following matrix interpretation satisfying 79.79/20.67 not(EDA) and not(IDA(1)). 79.79/20.67 79.79/20.67 [f[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [True] = [1] 79.79/20.67 79.79/20.67 [Nil] = [0] 79.79/20.67 79.79/20.67 [lt0](x1, x2) = [1] 79.79/20.67 79.79/20.67 [g](x1, x2) = [0] 79.79/20.67 79.79/20.67 [f](x1, x2) = [1] x1 + [1] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [0] 79.79/20.67 79.79/20.67 [Cons](x1, x2) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [number4](x1) = [1] x1 + [7] 79.79/20.67 79.79/20.67 [notEmpty](x1) = [4] 79.79/20.67 79.79/20.67 [goal](x1, x2) = [1] x1 + [1] x2 + [7] 79.79/20.67 79.79/20.67 [False] = [1] 79.79/20.67 79.79/20.67 The order satisfies the following ordering constraints: 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](True(), x', Cons(x, xs))] = [1] x' + [1] 79.79/20.67 >= [1] x' + [1] 79.79/20.67 = [f(x', xs)] 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](False(), Cons(x, xs), y)] = [1] xs + [1] x + [1] 79.79/20.67 >= [1] xs + [1] 79.79/20.67 = [f(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.67 79.79/20.67 [lt0(x, Nil())] = [1] 79.79/20.67 >= [1] 79.79/20.67 = [False()] 79.79/20.67 79.79/20.67 [lt0(Nil(), Cons(x', xs))] = [1] 79.79/20.67 >= [1] 79.79/20.67 = [True()] 79.79/20.67 79.79/20.67 [lt0(Cons(x', xs'), Cons(x, xs))] = [1] 79.79/20.67 >= [1] 79.79/20.67 = [lt0(xs', xs)] 79.79/20.67 79.79/20.67 [g(x, Nil())] = [0] 79.79/20.67 >= [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [g(x, Cons(x', xs))] = [0] 79.79/20.67 ? [1] 79.79/20.67 = [g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.67 79.79/20.67 [f(x, Nil())] = [1] x + [1] 79.79/20.67 > [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [f(x, Cons(x', xs))] = [1] x + [1] 79.79/20.67 >= [1] x + [1] 79.79/20.67 = [f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite](True(), x', Cons(x, xs))] = [1] 79.79/20.67 > [0] 79.79/20.67 = [g(x', xs)] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite](False(), Cons(x, xs), y)] = [1] 79.79/20.67 > [0] 79.79/20.67 = [g(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.67 79.79/20.67 [number4(n)] = [1] n + [7] 79.79/20.67 > [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [notEmpty(Nil())] = [4] 79.79/20.67 > [1] 79.79/20.67 = [False()] 79.79/20.67 79.79/20.67 [notEmpty(Cons(x, xs))] = [4] 79.79/20.67 > [1] 79.79/20.67 = [True()] 79.79/20.67 79.79/20.67 [goal(x, y)] = [1] x + [1] y + [7] 79.79/20.67 > [1] x + [1] 79.79/20.67 = [Cons(f(x, y), Cons(g(x, y), Nil()))] 79.79/20.67 79.79/20.67 79.79/20.67 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 79.79/20.67 79.79/20.67 We are left with following problem, upon which TcT provides the 79.79/20.67 certificate YES(O(1),O(n^1)). 79.79/20.67 79.79/20.67 Strict Trs: 79.79/20.67 { lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.67 , g(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , g(x, Cons(x', xs)) -> 79.79/20.67 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.67 , f(x, Cons(x', xs)) -> 79.79/20.67 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) } 79.79/20.67 Weak Trs: 79.79/20.67 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.67 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , lt0(x, Nil()) -> False() 79.79/20.67 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.67 , f(x, Nil()) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.67 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.67 g(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.67 , number4(n) -> 79.79/20.67 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.67 , notEmpty(Nil()) -> False() 79.79/20.67 , notEmpty(Cons(x, xs)) -> True() 79.79/20.67 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.67 Obligation: 79.79/20.67 innermost runtime complexity 79.79/20.67 Answer: 79.79/20.67 YES(O(1),O(n^1)) 79.79/20.67 79.79/20.67 The weightgap principle applies (using the following nonconstant 79.79/20.67 growth matrix-interpretation) 79.79/20.67 79.79/20.67 The following argument positions are usable: 79.79/20.67 Uargs(f[Ite][False][Ite]) = {1}, Uargs(g[Ite][False][Ite]) = {1}, 79.79/20.67 Uargs(Cons) = {1, 2} 79.79/20.67 79.79/20.67 TcT has computed the following matrix interpretation satisfying 79.79/20.67 not(EDA) and not(IDA(1)). 79.79/20.67 79.79/20.67 [f[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [True] = [1] 79.79/20.67 79.79/20.67 [Nil] = [0] 79.79/20.67 79.79/20.67 [lt0](x1, x2) = [1] 79.79/20.67 79.79/20.67 [g](x1, x2) = [1] 79.79/20.67 79.79/20.67 [f](x1, x2) = [1] x1 + [0] 79.79/20.67 79.79/20.67 [g[Ite][False][Ite]](x1, x2, x3) = [1] x1 + [0] 79.79/20.67 79.79/20.67 [Cons](x1, x2) = [1] x1 + [1] x2 + [0] 79.79/20.67 79.79/20.67 [number4](x1) = [1] x1 + [7] 79.79/20.67 79.79/20.67 [notEmpty](x1) = [4] 79.79/20.67 79.79/20.67 [goal](x1, x2) = [1] x1 + [1] x2 + [7] 79.79/20.67 79.79/20.67 [False] = [1] 79.79/20.67 79.79/20.67 The order satisfies the following ordering constraints: 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](True(), x', Cons(x, xs))] = [1] x' + [1] 79.79/20.67 > [1] x' + [0] 79.79/20.67 = [f(x', xs)] 79.79/20.67 79.79/20.67 [f[Ite][False][Ite](False(), Cons(x, xs), y)] = [1] xs + [1] x + [1] 79.79/20.67 > [1] xs + [0] 79.79/20.67 = [f(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.67 79.79/20.67 [lt0(x, Nil())] = [1] 79.79/20.67 >= [1] 79.79/20.67 = [False()] 79.79/20.67 79.79/20.67 [lt0(Nil(), Cons(x', xs))] = [1] 79.79/20.67 >= [1] 79.79/20.67 = [True()] 79.79/20.67 79.79/20.67 [lt0(Cons(x', xs'), Cons(x, xs))] = [1] 79.79/20.67 >= [1] 79.79/20.67 = [lt0(xs', xs)] 79.79/20.67 79.79/20.67 [g(x, Nil())] = [1] 79.79/20.67 > [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.67 [g(x, Cons(x', xs))] = [1] 79.79/20.67 >= [1] 79.79/20.67 = [g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.67 79.79/20.67 [f(x, Nil())] = [1] x + [0] 79.79/20.67 >= [0] 79.79/20.67 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.67 79.79/20.68 [f(x, Cons(x', xs))] = [1] x + [0] 79.79/20.68 ? [1] x + [1] 79.79/20.68 = [f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite](True(), x', Cons(x, xs))] = [1] 79.79/20.68 >= [1] 79.79/20.68 = [g(x', xs)] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite](False(), Cons(x, xs), y)] = [1] 79.79/20.68 >= [1] 79.79/20.68 = [g(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.68 79.79/20.68 [number4(n)] = [1] n + [7] 79.79/20.68 > [0] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [notEmpty(Nil())] = [4] 79.79/20.68 > [1] 79.79/20.68 = [False()] 79.79/20.68 79.79/20.68 [notEmpty(Cons(x, xs))] = [4] 79.79/20.68 > [1] 79.79/20.68 = [True()] 79.79/20.68 79.79/20.68 [goal(x, y)] = [1] x + [1] y + [7] 79.79/20.68 > [1] x + [1] 79.79/20.68 = [Cons(f(x, y), Cons(g(x, y), Nil()))] 79.79/20.68 79.79/20.68 79.79/20.68 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 79.79/20.68 79.79/20.68 We are left with following problem, upon which TcT provides the 79.79/20.68 certificate YES(O(1),O(n^1)). 79.79/20.68 79.79/20.68 Strict Trs: 79.79/20.68 { lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.68 , g(x, Cons(x', xs)) -> 79.79/20.68 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.68 , f(x, Cons(x', xs)) -> 79.79/20.68 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) } 79.79/20.68 Weak Trs: 79.79/20.68 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.68 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.68 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.68 , lt0(x, Nil()) -> False() 79.79/20.68 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.68 , g(x, Nil()) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , f(x, Nil()) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.68 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.68 g(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.68 , number4(n) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , notEmpty(Nil()) -> False() 79.79/20.68 , notEmpty(Cons(x, xs)) -> True() 79.79/20.68 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.68 Obligation: 79.79/20.68 innermost runtime complexity 79.79/20.68 Answer: 79.79/20.68 YES(O(1),O(n^1)) 79.79/20.68 79.79/20.68 We use the processor 'matrix interpretation of dimension 2' to 79.79/20.68 orient following rules strictly. 79.79/20.68 79.79/20.68 Trs: 79.79/20.68 { g(x, Cons(x', xs)) -> 79.79/20.68 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) } 79.79/20.68 79.79/20.68 The induced complexity on above rules (modulo remaining rules) is 79.79/20.68 YES(?,O(n^1)) . These rules are moved into the corresponding weak 79.79/20.68 component(s). 79.79/20.68 79.79/20.68 Sub-proof: 79.79/20.68 ---------- 79.79/20.68 The following argument positions are usable: 79.79/20.68 Uargs(f[Ite][False][Ite]) = {1}, Uargs(g[Ite][False][Ite]) = {1}, 79.79/20.68 Uargs(Cons) = {1, 2} 79.79/20.68 79.79/20.68 TcT has computed the following constructor-based matrix 79.79/20.68 interpretation satisfying not(EDA) and not(IDA(1)). 79.79/20.68 79.79/20.68 [f[Ite][False][Ite]](x1, x2, x3) = [2 0] x1 + [3] 79.79/20.68 [0 0] [1] 79.79/20.68 79.79/20.68 [True] = [0] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 [Nil] = [0] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 [lt0](x1, x2) = [0] 79.79/20.68 [4] 79.79/20.68 79.79/20.68 [g](x1, x2) = [4 4] x1 + [1 1] x2 + [3] 79.79/20.68 [0 0] [0 0] [4] 79.79/20.68 79.79/20.68 [f](x1, x2) = [3] 79.79/20.68 [1] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite]](x1, x2, x3) = [4 0] x1 + [4 4] x2 + [1 79.79/20.68 3] x3 + [0] 79.79/20.68 [0 0] [0 0] [0 79.79/20.68 0] [4] 79.79/20.68 79.79/20.68 [Cons](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 79.79/20.68 [0 0] [0 0] [1] 79.79/20.68 79.79/20.68 [number4](x1) = [7 7] x1 + [7] 79.79/20.68 [0 0] [7] 79.79/20.68 79.79/20.68 [notEmpty](x1) = [0] 79.79/20.68 [4] 79.79/20.68 79.79/20.68 [goal](x1, x2) = [7 7] x1 + [7 7] x2 + [7] 79.79/20.68 [7 7] [7 7] [7] 79.79/20.68 79.79/20.68 [False] = [0] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 The order satisfies the following ordering constraints: 79.79/20.68 79.79/20.68 [f[Ite][False][Ite](True(), x', Cons(x, xs))] = [3] 79.79/20.68 [1] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [f(x', xs)] 79.79/20.68 79.79/20.68 [f[Ite][False][Ite](False(), Cons(x, xs), y)] = [3] 79.79/20.68 [1] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [f(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.68 79.79/20.68 [lt0(x, Nil())] = [0] 79.79/20.68 [4] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [False()] 79.79/20.68 79.79/20.68 [lt0(Nil(), Cons(x', xs))] = [0] 79.79/20.68 [4] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [True()] 79.79/20.68 79.79/20.68 [lt0(Cons(x', xs'), Cons(x, xs))] = [0] 79.79/20.68 [4] 79.79/20.68 >= [0] 79.79/20.68 [4] 79.79/20.68 = [lt0(xs', xs)] 79.79/20.68 79.79/20.68 [g(x, Nil())] = [4 4] x + [3] 79.79/20.68 [0 0] [4] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [g(x, Cons(x', xs))] = [1 0] x' + [1 1] xs + [4 4] x + [4] 79.79/20.68 [0 0] [0 0] [0 0] [4] 79.79/20.68 > [1 0] x' + [1 1] xs + [4 4] x + [3] 79.79/20.68 [0 0] [0 0] [0 0] [4] 79.79/20.68 = [g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.68 79.79/20.68 [f(x, Nil())] = [3] 79.79/20.68 [1] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [f(x, Cons(x', xs))] = [3] 79.79/20.68 [1] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite](True(), x', Cons(x, xs))] = [4 4] x' + [1 1] xs + [1 0] x + [3] 79.79/20.68 [0 0] [0 0] [0 0] [4] 79.79/20.68 >= [4 4] x' + [1 1] xs + [3] 79.79/20.68 [0 0] [0 0] [4] 79.79/20.68 = [g(x', xs)] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite](False(), Cons(x, xs), y)] = [4 4] xs + [4 0] x + [1 3] y + [4] 79.79/20.68 [0 0] [0 0] [0 0] [4] 79.79/20.68 >= [4 4] xs + [1 1] y + [4] 79.79/20.68 [0 0] [0 0] [4] 79.79/20.68 = [g(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.68 79.79/20.68 [number4(n)] = [7 7] n + [7] 79.79/20.68 [0 0] [7] 79.79/20.68 > [3] 79.79/20.68 [1] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [notEmpty(Nil())] = [0] 79.79/20.68 [4] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [False()] 79.79/20.68 79.79/20.68 [notEmpty(Cons(x, xs))] = [0] 79.79/20.68 [4] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [True()] 79.79/20.68 79.79/20.68 [goal(x, y)] = [7 7] x + [7 7] y + [7] 79.79/20.68 [7 7] [7 7] [7] 79.79/20.68 >= [4 4] x + [1 1] y + [7] 79.79/20.68 [0 0] [0 0] [1] 79.79/20.68 = [Cons(f(x, y), Cons(g(x, y), Nil()))] 79.79/20.68 79.79/20.68 79.79/20.68 We return to the main proof. 79.79/20.68 79.79/20.68 We are left with following problem, upon which TcT provides the 79.79/20.68 certificate YES(O(1),O(n^1)). 79.79/20.68 79.79/20.68 Strict Trs: 79.79/20.68 { lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.68 , f(x, Cons(x', xs)) -> 79.79/20.68 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) } 79.79/20.68 Weak Trs: 79.79/20.68 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.68 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.68 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.68 , lt0(x, Nil()) -> False() 79.79/20.68 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.68 , g(x, Nil()) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , g(x, Cons(x', xs)) -> 79.79/20.68 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.68 , f(x, Nil()) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.68 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.68 g(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.68 , number4(n) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , notEmpty(Nil()) -> False() 79.79/20.68 , notEmpty(Cons(x, xs)) -> True() 79.79/20.68 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.68 Obligation: 79.79/20.68 innermost runtime complexity 79.79/20.68 Answer: 79.79/20.68 YES(O(1),O(n^1)) 79.79/20.68 79.79/20.68 We use the processor 'matrix interpretation of dimension 2' to 79.79/20.68 orient following rules strictly. 79.79/20.68 79.79/20.68 Trs: { lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) } 79.79/20.68 79.79/20.68 The induced complexity on above rules (modulo remaining rules) is 79.79/20.68 YES(?,O(n^1)) . These rules are moved into the corresponding weak 79.79/20.68 component(s). 79.79/20.68 79.79/20.68 Sub-proof: 79.79/20.68 ---------- 79.79/20.68 The following argument positions are usable: 79.79/20.68 Uargs(f[Ite][False][Ite]) = {1}, Uargs(g[Ite][False][Ite]) = {1}, 79.79/20.68 Uargs(Cons) = {1, 2} 79.79/20.68 79.79/20.68 TcT has computed the following constructor-based matrix 79.79/20.68 interpretation satisfying not(EDA) and not(IDA(1)). 79.79/20.68 79.79/20.68 [f[Ite][False][Ite]](x1, x2, x3) = [1 0] x1 + [0 1] x2 + [0] 79.79/20.68 [0 0] [0 0] [4] 79.79/20.68 79.79/20.68 [True] = [1] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 [Nil] = [0] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 [lt0](x1, x2) = [0 1] x2 + [0] 79.79/20.68 [0 0] [4] 79.79/20.68 79.79/20.68 [g](x1, x2) = [0 2] x1 + [1] 79.79/20.68 [0 2] [6] 79.79/20.68 79.79/20.68 [f](x1, x2) = [0 1] x1 + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite]](x1, x2, x3) = [1 0] x1 + [0 2] x2 + [0] 79.79/20.68 [0 0] [0 2] [6] 79.79/20.68 79.79/20.68 [Cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 79.79/20.68 [0 0] [0 1] [1] 79.79/20.68 79.79/20.68 [number4](x1) = [7 7] x1 + [7] 79.79/20.68 [0 0] [7] 79.79/20.68 79.79/20.68 [notEmpty](x1) = [0 4] x1 + [0] 79.79/20.68 [0 0] [4] 79.79/20.68 79.79/20.68 [goal](x1, x2) = [7 7] x1 + [7 7] x2 + [7] 79.79/20.68 [7 7] [7 7] [7] 79.79/20.68 79.79/20.68 [False] = [0] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 The order satisfies the following ordering constraints: 79.79/20.68 79.79/20.68 [f[Ite][False][Ite](True(), x', Cons(x, xs))] = [0 1] x' + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 >= [0 1] x' + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 = [f(x', xs)] 79.79/20.68 79.79/20.68 [f[Ite][False][Ite](False(), Cons(x, xs), y)] = [0 1] xs + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 >= [0 1] xs + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 = [f(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.68 79.79/20.68 [lt0(x, Nil())] = [0] 79.79/20.68 [4] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [False()] 79.79/20.68 79.79/20.68 [lt0(Nil(), Cons(x', xs))] = [0 1] xs + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 >= [1] 79.79/20.68 [0] 79.79/20.68 = [True()] 79.79/20.68 79.79/20.68 [lt0(Cons(x', xs'), Cons(x, xs))] = [0 1] xs + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 > [0 1] xs + [0] 79.79/20.68 [0 0] [4] 79.79/20.68 = [lt0(xs', xs)] 79.79/20.68 79.79/20.68 [g(x, Nil())] = [0 2] x + [1] 79.79/20.68 [0 2] [6] 79.79/20.68 > [0] 79.79/20.68 [4] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [g(x, Cons(x', xs))] = [0 2] x + [1] 79.79/20.68 [0 2] [6] 79.79/20.68 >= [0 2] x + [1] 79.79/20.68 [0 2] [6] 79.79/20.68 = [g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.68 79.79/20.68 [f(x, Nil())] = [0 1] x + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 > [0] 79.79/20.68 [4] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [f(x, Cons(x', xs))] = [0 1] x + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 >= [0 1] x + [1] 79.79/20.68 [0 0] [4] 79.79/20.68 = [f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite](True(), x', Cons(x, xs))] = [0 2] x' + [1] 79.79/20.68 [0 2] [6] 79.79/20.68 >= [0 2] x' + [1] 79.79/20.68 [0 2] [6] 79.79/20.68 = [g(x', xs)] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite](False(), Cons(x, xs), y)] = [0 2] xs + [2] 79.79/20.68 [0 2] [8] 79.79/20.68 > [0 2] xs + [1] 79.79/20.68 [0 2] [6] 79.79/20.68 = [g(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.68 79.79/20.68 [number4(n)] = [7 7] n + [7] 79.79/20.68 [0 0] [7] 79.79/20.68 > [0] 79.79/20.68 [4] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [notEmpty(Nil())] = [0] 79.79/20.68 [4] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [False()] 79.79/20.68 79.79/20.68 [notEmpty(Cons(x, xs))] = [0 4] xs + [4] 79.79/20.68 [0 0] [4] 79.79/20.68 > [1] 79.79/20.68 [0] 79.79/20.68 = [True()] 79.79/20.68 79.79/20.68 [goal(x, y)] = [7 7] x + [7 7] y + [7] 79.79/20.68 [7 7] [7 7] [7] 79.79/20.68 > [0 3] x + [2] 79.79/20.68 [0 0] [2] 79.79/20.68 = [Cons(f(x, y), Cons(g(x, y), Nil()))] 79.79/20.68 79.79/20.68 79.79/20.68 We return to the main proof. 79.79/20.68 79.79/20.68 We are left with following problem, upon which TcT provides the 79.79/20.68 certificate YES(O(1),O(n^1)). 79.79/20.68 79.79/20.68 Strict Trs: 79.79/20.68 { f(x, Cons(x', xs)) -> 79.79/20.68 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) } 79.79/20.68 Weak Trs: 79.79/20.68 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.68 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.68 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.68 , lt0(x, Nil()) -> False() 79.79/20.68 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.68 , lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.68 , g(x, Nil()) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , g(x, Cons(x', xs)) -> 79.79/20.68 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.68 , f(x, Nil()) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.68 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.68 g(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.68 , number4(n) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , notEmpty(Nil()) -> False() 79.79/20.68 , notEmpty(Cons(x, xs)) -> True() 79.79/20.68 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.68 Obligation: 79.79/20.68 innermost runtime complexity 79.79/20.68 Answer: 79.79/20.68 YES(O(1),O(n^1)) 79.79/20.68 79.79/20.68 We use the processor 'matrix interpretation of dimension 2' to 79.79/20.68 orient following rules strictly. 79.79/20.68 79.79/20.68 Trs: 79.79/20.68 { f(x, Cons(x', xs)) -> 79.79/20.68 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) } 79.79/20.68 79.79/20.68 The induced complexity on above rules (modulo remaining rules) is 79.79/20.68 YES(?,O(n^1)) . These rules are moved into the corresponding weak 79.79/20.68 component(s). 79.79/20.68 79.79/20.68 Sub-proof: 79.79/20.68 ---------- 79.79/20.68 The following argument positions are usable: 79.79/20.68 Uargs(f[Ite][False][Ite]) = {1}, Uargs(g[Ite][False][Ite]) = {1}, 79.79/20.68 Uargs(Cons) = {1, 2} 79.79/20.68 79.79/20.68 TcT has computed the following constructor-based matrix 79.79/20.68 interpretation satisfying not(EDA) and not(IDA(1)). 79.79/20.68 79.79/20.68 [f[Ite][False][Ite]](x1, x2, x3) = [4 0] x1 + [4 4] x2 + [1 79.79/20.68 1] x3 + [2] 79.79/20.68 [0 0] [4 4] [4 79.79/20.68 4] [0] 79.79/20.68 79.79/20.68 [True] = [0] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 [Nil] = [0] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 [lt0](x1, x2) = [0] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 [g](x1, x2) = [3] 79.79/20.68 [1] 79.79/20.68 79.79/20.68 [f](x1, x2) = [4 4] x1 + [1 1] x2 + [3] 79.79/20.68 [4 4] [4 1] [3] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite]](x1, x2, x3) = [1 0] x1 + [3] 79.79/20.68 [0 0] [1] 79.79/20.68 79.79/20.68 [Cons](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 79.79/20.68 [0 0] [0 0] [1] 79.79/20.68 79.79/20.68 [number4](x1) = [7 7] x1 + [7] 79.79/20.68 [0 0] [7] 79.79/20.68 79.79/20.68 [notEmpty](x1) = [0] 79.79/20.68 [4] 79.79/20.68 79.79/20.68 [goal](x1, x2) = [7 7] x1 + [7 7] x2 + [7] 79.79/20.68 [7 7] [7 7] [7] 79.79/20.68 79.79/20.68 [False] = [0] 79.79/20.68 [0] 79.79/20.68 79.79/20.68 The order satisfies the following ordering constraints: 79.79/20.68 79.79/20.68 [f[Ite][False][Ite](True(), x', Cons(x, xs))] = [4 4] x' + [1 1] xs + [1 0] x + [3] 79.79/20.68 [4 4] [4 4] [4 0] [4] 79.79/20.68 >= [4 4] x' + [1 1] xs + [3] 79.79/20.68 [4 4] [4 1] [3] 79.79/20.68 = [f(x', xs)] 79.79/20.68 79.79/20.68 [f[Ite][False][Ite](False(), Cons(x, xs), y)] = [4 4] xs + [4 0] x + [1 1] y + [6] 79.79/20.68 [4 4] [4 0] [4 4] [4] 79.79/20.68 > [4 4] xs + [1 1] y + [4] 79.79/20.68 [4 4] [4 4] [4] 79.79/20.68 = [f(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.68 79.79/20.68 [lt0(x, Nil())] = [0] 79.79/20.68 [0] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [False()] 79.79/20.68 79.79/20.68 [lt0(Nil(), Cons(x', xs))] = [0] 79.79/20.68 [0] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [True()] 79.79/20.68 79.79/20.68 [lt0(Cons(x', xs'), Cons(x, xs))] = [0] 79.79/20.68 [0] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [lt0(xs', xs)] 79.79/20.68 79.79/20.68 [g(x, Nil())] = [3] 79.79/20.68 [1] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [g(x, Cons(x', xs))] = [3] 79.79/20.68 [1] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.68 79.79/20.68 [f(x, Nil())] = [4 4] x + [3] 79.79/20.68 [4 4] [3] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [f(x, Cons(x', xs))] = [1 0] x' + [1 1] xs + [4 4] x + [4] 79.79/20.68 [4 0] [4 4] [4 4] [4] 79.79/20.68 > [1 0] x' + [1 1] xs + [4 4] x + [3] 79.79/20.68 [4 0] [4 4] [4 4] [4] 79.79/20.68 = [f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs))] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite](True(), x', Cons(x, xs))] = [3] 79.79/20.68 [1] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [g(x', xs)] 79.79/20.68 79.79/20.68 [g[Ite][False][Ite](False(), Cons(x, xs), y)] = [3] 79.79/20.68 [1] 79.79/20.68 >= [3] 79.79/20.68 [1] 79.79/20.68 = [g(xs, Cons(Cons(Nil(), Nil()), y))] 79.79/20.68 79.79/20.68 [number4(n)] = [7 7] n + [7] 79.79/20.68 [0 0] [7] 79.79/20.68 > [3] 79.79/20.68 [1] 79.79/20.68 = [Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil()))))] 79.79/20.68 79.79/20.68 [notEmpty(Nil())] = [0] 79.79/20.68 [4] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [False()] 79.79/20.68 79.79/20.68 [notEmpty(Cons(x, xs))] = [0] 79.79/20.68 [4] 79.79/20.68 >= [0] 79.79/20.68 [0] 79.79/20.68 = [True()] 79.79/20.68 79.79/20.68 [goal(x, y)] = [7 7] x + [7 7] y + [7] 79.79/20.68 [7 7] [7 7] [7] 79.79/20.68 >= [4 4] x + [1 1] y + [7] 79.79/20.68 [0 0] [0 0] [1] 79.79/20.68 = [Cons(f(x, y), Cons(g(x, y), Nil()))] 79.79/20.68 79.79/20.68 79.79/20.68 We return to the main proof. 79.79/20.68 79.79/20.68 We are left with following problem, upon which TcT provides the 79.79/20.68 certificate YES(O(1),O(1)). 79.79/20.68 79.79/20.68 Weak Trs: 79.79/20.68 { f[Ite][False][Ite](True(), x', Cons(x, xs)) -> f(x', xs) 79.79/20.68 , f[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.68 f(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.68 , lt0(x, Nil()) -> False() 79.79/20.68 , lt0(Nil(), Cons(x', xs)) -> True() 79.79/20.68 , lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs) 79.79/20.68 , g(x, Nil()) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , g(x, Cons(x', xs)) -> 79.79/20.68 g[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.68 , f(x, Nil()) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , f(x, Cons(x', xs)) -> 79.79/20.68 f[Ite][False][Ite](lt0(x, Cons(Nil(), Nil())), x, Cons(x', xs)) 79.79/20.68 , g[Ite][False][Ite](True(), x', Cons(x, xs)) -> g(x', xs) 79.79/20.68 , g[Ite][False][Ite](False(), Cons(x, xs), y) -> 79.79/20.68 g(xs, Cons(Cons(Nil(), Nil()), y)) 79.79/20.68 , number4(n) -> 79.79/20.68 Cons(Nil(), Cons(Nil(), Cons(Nil(), Cons(Nil(), Nil())))) 79.79/20.68 , notEmpty(Nil()) -> False() 79.79/20.68 , notEmpty(Cons(x, xs)) -> True() 79.79/20.68 , goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil())) } 79.79/20.68 Obligation: 79.79/20.68 innermost runtime complexity 79.79/20.68 Answer: 79.79/20.68 YES(O(1),O(1)) 79.79/20.68 79.79/20.68 Empty rules are trivially bounded 79.79/20.68 79.79/20.68 Hurray, we answered YES(O(1),O(n^1)) 80.06/20.77 EOF